The MicroLogix 1100 will show you the IP address using the keypad/LCD, but it won't let you configure them.
See the "Using the LCD" section in the 1763-UM001 User Manual to see how to navigate the LCD screen to the Advanced Settings section, where you can see the IP address (or the DHCP/BOOTP settings).
If there is no IP address assigned, the Advanced Settings will show the MAC ID only. If there is an IP address assigned, it will show both the MAC ID and the IP Address.
You can verify IP connectivity using PING or by using HTTP. Type the IP address of the controller into a web browser and you should see the diagnostic web pages.
I don't think any versions of RSLinx Classic are supported in Windows 8, but version 2.57 predates Windows 7. See if you can download version 3.6, as RSLinx Classic Lite is always available free of charge from the main RA software download site.
On the serial port, you can use the DCOMM button to force the port to the default settings.
If you're comfortable with using a serial terminal like RealTerm, you could send the hex bytes "10 05" to the serial port at 19200 baud, 8 data bits, no parity, and you should get the MicroLogix port to echo "10 05" back to you. That will at least verify that the serial port on the controller, the serial cable, and your USB/RS232 converter are functional.